[petalas]

Summary

Remove the odd divisor from the five-product Toom-3 interpolation in Gonzalo Mier's PR3974, with an explicit preserving/restoring boundary. Claim type: scoped algebraic proof and finite construction certificate. No new kappa or selected-frontier change.

old evaluation: 0, 1, -1, 2, infinity; interpolation includes division by 3
new evaluation: 0, 1,  i, 1+i, infinity; inverse entries in (1+i)^(-1) Z[i]

encode five dirty outputs by V
add the five evaluated products directly into those outputs
restore the two clean evaluation fields after each use
apply V^(-1)

The new matrix has determinant 2. Denominator height one in 1+i is optimal among integral point-evaluation schemes with Gaussian-dyadic linear interpolation. The literal word preserves six inputs and arbitrary dirty outputs, restoring two initially zero fields. It pays five PRODUCT calls, sixty coefficient ADDs and four SCALEs. The ADD coefficients are explicit charged macros; no total scalar or physical speedup is claimed. The two divisions by associates of 1+i are exact on every valid Gaussian-integer input. The usual bilinear lower bound of five products is attained.

The new source-relative module and proof were produced by Nick Petalas. Credit Gonzalo Mier for the screen and embedding question, Chris R / Clear1nsight for PR3992, Josie Elliston, Arham Tahir and the retained PR1094/Clifford lineage. Toom-Cook evaluation, Lagrange interpolation and tensor flattening are classical; global literature priority for complex Toom formulas is unclaimed. Predecessor scientific computations are cited, not replayed or audited.

Evidence

• Before: the retained five-product screen uses an odd-denominator interpolation and supplies no complete restoring product boundary.
• After: a seven-file isolated package supplies the exact interpolation, complete 69-instruction word, written sharp denominator obstruction, and separate schoolbook/rational-matrix oracles.
• Both warning-strict CPython 3.12.15 modes pass all 25 inverse entries, five encoder and five decoder columns, 46 tensor/dirty-output columns, 988 complete endpoint fixtures, every one of 69 instruction omissions, five invalid entrances and an inexact-division control. Complete inverse endpoints are also checked on the 729 real cube and 257 complex fixtures. The two large fixtures include the maximum 4096-bit entrance.
• Both registered commands pass locally in under 0.1 seconds each. Metadata, all seven committed files and declared source ancestry pass. Official export integrity and input coverage pass; both read-only, Git-free, empty-PATH replays pass with all eight export files unchanged in bytes and modes. The verifier generates its receipt from source hashes and writes no package files.

From the repository root, with the CPython 3.12 standard library:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/gaussian-dyadic-toom3/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/gaussian-dyadic-toom3/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/gaussian-dyadic-toom3
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/gaussian-dyadic-toom3 --check-source-history

Another CPython 3.12 path may be substituted. Expected markers are PASS_GAUSSIAN_DYADIC_TOOM3, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. The written proofs supply the universal quantifiers; hosted execution of these finite checks is not a full theorem proof.

Limitations and unrun obligations

This module needs a genuine convolution consumer, compatible frames and two initially clean fields. Their allocation, coefficient access, native PRODUCT implementation, routing, precision, stock and complete signed recursive/outer invoice are unpriced. PR3992's same-width obstruction for the first historical nine-climb group remains: changing interpolation supplies no extra information channel. No actual selected-provider sub-tensor or supplier admission is claimed. All five selected assumptions retain their status.

The API admits at most 4096 bits per component; outputs may exceed that entrance cap. The proof has no bit cap. Arbitrary dirty evaluation scratch, non-evaluation interpolation optimality and arbitrary nonlinear product lower bounds are unproved. No selected/native cold replay, Lean, Docker, independent human acceptance or formal proof-kernel review is supplied.

Open next questions

1. Bind a genuine bilinear convolution consumer and price its full input/output, frame, native and signed recursive contract. The missing physical caller and invoice block an algorithmic gain.
2. Determine whether arbitrary dirty evaluation scratch, or fully charged clean-field admission, preserves the five-call advantage. Start with that exact workspace contract and charge every additional product and field pass.

Related open work

Completion refresh at 2026-11-03 11:27 UTC inspected this account's 12 open PRs created since the advance claim. None of the other 11 studies the five-point Toom-3 interpolation boundary, so no numerical comparison is applicable. PR3974 and PR3992 remain credited predecessors on main. The live Toom search before publication found no other open PR on this boundary. Arbitrary diffs and unpublished work remain overlap limits.

CI notes

Both hosted checks, plan and replay (research/gaussian-dyadic-toom3), passed at c069c79e81dd97296e2cae07c7d627c26ab19947 in run 41626433244. The completed checks were read once; run metadata confirms completed success and the matching head. Hosted execution certifies the registered bounded replay, not independent acceptance of the universal proof. All local checks listed above also passed. Advance claim.

Merge Danger

Door: two-way. The change adds only research/gaussian-dyadic-toom3/ and its opt-in replay manifest.

Blast Radius: isolated. Existing sources, credits, selected records and shared code are unchanged.